mathematics An implementation of
ordinals in
set theory (e.g.
Zermelo Fränkel set theory or
ZFC). The von Neumann ordinal alpha is the
well-ordered set containing just the ordinals "shorter" than alpha.
"Reasonable" set theories (like ZF) include Mostowski's Collapsing Theorem: any
well-ordered set is
isomorphic to a von Neumann ordinal. In really screwy theories (e.g. NFU -- New Foundations with Urelemente) this theorem is false.
The finite von Neumann ordinals are the von Neumann integers.
(1995-03-30)